CHAP. Hi] PARTITIONS. 137 



sented by n units in a row), the sums of the columns reproduce the original 

 partition. Thus 4 3 2 1 is a self-conjugate partition of 10. Evidently 



4321 



every such array contains a central square of q 2 units (4 in the diagram), 

 where q is odd or even, according as the partitioned number n is odd or 

 even, since of the n q 2 units outside the square half are at the right and 

 half below the square. The partition remains self-conjugate under any 

 rearrangement of the (n # 2 )/2 units to the right, provided those below 

 be arranged symmetrically. The number {%(n q*}; q} of such rearrange- 

 ments is the number of ways of dividing %(n g 2 ) into q or fewer parts. 

 In the above diagram we may replace the double row of three dots to the 

 right of the square by a single row of three dots and derive the only other 

 self-conjugate partition of 10. In general, the number of self-conjugate 

 partitions of n is 2{|(n g 2 ); q}, summed for all odd or all even integers 

 q < ^, according as n is odd or even. 



J. J. Sylvester 112 noted that Durfee's 111 theorem may be expressed in the 

 following form: The number of self-conjugate partitions of n (or of sym- 

 metrical partition graphs for n) is the coefficient of x n in 



and hence is the number of partitions of n into unrepeated odd integers. 

 He gave a modification of Franklin's 105 proof of (3). 



Sylvester 113 proved Brioschi's 47 formula Z = ^(r}z r . 



Sylvester 114 proved by use of the binary scale Euler's theorem that the 

 number of partitions of n into odd parts equals the number of its partitions 

 into distinct parts [Glaisher 86 ]]. Of graphical methods in partitions, he 

 called Ferrers' 35 method transversion and Durfee's 111 method apocopation. 

 He gave a graphical proof of Euler's (3). 



F. Franklin 115 noted that, since the number (w; i, j] of ways w can be 

 partitioned into i or fewer parts ^ j is the coefficient of a j x w in the develop- 

 ment of the reciprocal of (1 a)(l ax) (1 ax*), the coefficient of a j 

 in its development in ascending powers of a is the generating function F 

 in which the coefficient of x w is (w; i, j). To obtain F directly, note that 

 the number of ways of forming w with i or fewer parts of which at least one 

 is a number > j, say j + k, equals the number of ways of forming 

 w (j + k) with i 1 or fewer parts; the number of partitions in which 



112 Johns Hopkins Univ. Circ., 2, 1882-3, 23-24, 42-4; Coll. Math. Papers, III, 661-671. 

 113 Ibid., 2, 1883, 46; Coll. Math. Papers, III, 677-9; Amer. Jour. Math., 5, 1882, 271-2; 

 Coll. Math. Papers, IV, 21-23. 



114 Ibid., 70-71; Coll. Math. Papers, III, 680-6. Cf. Coll. Math. Papers, IV, 13-18. 



115 Ibid., 72 (in full). 



